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Linear measurement methods; sag correction

Short answer

Linear measurement methods include pacing, passometer, odometer, chaining, taping, tacheometry, subtense bar, EDM and GNSS. Sag correction Cₛ = −W²l/(24P²) is negative (subtracted) when a tape standardised on the flat is used suspended, and positive (added) when standardised in catenary and used on the flat.

What are the methods of linear measurements? Explain about sag correction. When is the sag correction considered positive and negative?

Methods of linear measurement

  1. Pacing: counting the paces and multiplying by the average pace length (rough, accuracy about 1/100).
  2. Passometer/pedometer and odometer or measuring wheel: record the steps or the revolutions of a wheel (rough).
  3. Chaining or taping: direct measurement by a chain, steel tape or invar tape (ordinary to high accuracy, 1/1000 to 1/10,000 or better).
  4. Tacheometry: the stadia hairs of a theodolite give the distance from readings on a staff.
  5. Subtense bar method: a bar of known length subtends a small angle at the observer.
  6. EDM/total station: distance from the time or phase of electromagnetic waves.
  7. GNSS/GPS and photogrammetry: distances from coordinates.

Sag correction

When a tape is suspended between two supports, it hangs in the form of a catenary (approximately a parabola), and the length measured along the curve is longer than the horizontal chord. For a span of length ll in which the tape weighs WW (weight of the tape in that span, W=w lW = w\,l), under a pull PP:

Cs=−W2 l24 P2 per spanC_s = -\frac{W^{2}\,l}{24\,P^{2}} \text{ per span}

or, in terms of the weight per metre ww and the total length LL in nn equal spans:

Cs=−n w2 l324 P2=−L w2 l224 P2C_s = -\frac{n\,w^{2}\,l^{3}}{24\,P^{2}} = -\frac{L\,w^{2}\,l^{2}}{24\,P^{2}}

Sign of sag correction

  • Negative when the tape was standardised on the flat (fully supported) and is used suspended (in catenary), because the sagging tape reads more than the horizontal distance. The correction is subtracted.
  • Positive when the tape was standardised in catenary and is used on the flat (fully supported), because the sag effect included in its standard length must be removed. In this case it is added.
  • No correction is required if the standardisation and the use are under the same conditions, with the same pull and span.

Sag can be reduced by applying a higher pull or by shortening the spans. The pull at which the positive pull correction exactly balances the negative sag correction is called the normal tension; at this pull the two effects cancel.

✓ Answer checked by Claude Opus 5.5 (high effort)

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